DR. BABASAHEB AMBEDKAR MARATHWADA UNIVERSITY, AURANGABAD DEPARTMENT OF MATHEMATICS

Size: px

Start display at page:

Download "DR. BABASAHEB AMBEDKAR MARATHWADA UNIVERSITY, AURANGABAD DEPARTMENT OF MATHEMATICS"

Arleen Skinner
5 years ago
Views:

1 1 DR. BABASAHEB AMBEDKAR MARATHWADA UNIVERSITY, AURANGABAD DEPARTMENT OF MATHEMATICS Syllabus for M.A. / M. Sc. (Mathematics) Semester I, II, III, and IV Under Academic Flexibility of the Department W.E.F. JUNE 2010 The M. A. / M. Sc. (Mathematics) course consists of four semesters. In Semesters I and II a student has to study four compulsory and one optional papers. In Semesters III and IV he/she has to study two compulsory and three optional papers. Unitwise distribution of the syllabus for the papers currently taught is given. For other papers it will be given as and when the papers are introduced. SEMESTER- I Paper I (A) - Advanced Abstract Algebra -I Paper II (A) - Real Analysis -I Paper III (A) - Topology -I Paper IV (A) - Complex Analysis -I Optional Paper (Any one of the following) Paper V- a (A) - Differential Equations -I Paper V- b (A) - Advanced Discrete Mathematics -I Paper V- c (A) - Differential Geometry of Manifolds -I. SEMESTER -II Paper I (B) - Advanced Abstract Algebra -II Paper II (B) - Real Analysis -II Paper III (B) - Topology -II Paper IV (B) - Complex Analysis -II Optional Paper (Any one of the following) Paper V -a(b) - Differential Equations -II Paper V -b (B) - Advanced Discrete Mathematics -II Paper V -c (B) - Differential Geometry of Manifolds -II.

2 2 SEMESTER III Paper VI (A) - Functional Analysis - I Paper VII (A) - Partial Differential Equations Optional Papers (Any three of the following) Paper VIII (A) - Programming in C Theory and practical- I Paper IX (A) - Fluid Mechanics -I Paper X (A) - Integral Equations Paper XI (A) - Numerical Analysis -I Paper XII (A) - Lattice Theory -I Paper XIII (A) - Advanced Functional Analysis -I Paper XIV (A) - Advanced Theory of Partial Differential Equations -I Paper XV (A) - Theory of Ordinary Differential equations -I Paper XVI (A) - Difference Equations -I Paper XVII (A) - Computational Fluid Dynamics -I Paper XVIII (A) - Algebraic Coding Theory -I Paper XIX (A) - Algebraic Topology -I Paper XX (A) - Operations Research -I Paper XXI (A) - Banach Algebra -I Paper XXII (A) - Wavelets -I Paper XXIII (A) - Fundamentals of Applied Functional Analysis -I Paper XXIV (A) - Algebraic Number Theory -I Paper XXV (A) - Combinatorics -I Paper XXVI (A) - Reaction diffusion theory - I SEMESTER IV Paper VI (B) - Functional Analysis - II Paper VII (B) - Mechanics. Optional Papers (Any three of the following) Paper VIII (B) - MATLAB Programming Paper IX (B) - Fluid Mechanics -II Paper X (B) - Boundary Value Problems

3 Paper XI (B) - Fuzzy Mathematics Paper XII (B) - Linear Algebra Paper XIII (B) - Advanced Functional Analysis -II Paper XIV (B) - Advanced Theory of Partial Differential Equations -II Paper XV (B) - Theory of Ordinary Differential equations -II Paper XVI (B) - Difference Equations -II Paper XVII (B) - Computational Fluid dynamics -II Paper XVIII (B) - Algebraic Coding Theory -II Paper XIX (B) - Algebraic Topology -II Paper XX (B) - Operations Research -II Paper XXI (B) - Banach Algebra -II Paper XXII (B) - Wavelets -II Paper XXIII (B) - Fundamentals of Applied Functional Analysis -II Paper XXIV (B) - Algebraic Number Theory -II Paper XXV (B) - Combinatorics -II Paper XXVI (B) - Reaction diffusion theory - II

4 4 Semester I Paper- I (A) - Advanced Abstract Algebra- I Unit- I Preliminaries and related concepts from algebra and field theory. Unit- II Irreducible polynomials, Eisenstein criterion, adjunction of roots, algebraic extension of a field, algebraically closed field and related results. Unit- III Splitting fields, normal extensions, multiple roots, finite fields and separable extensions and their properties. ( 15 lectures) Unit IV Galois theory, automosphism groups and fixed fields, fundamental theorem of Galois theory, fundamental theorem of algebra. Unit- V Applications of Galois theory to classical problems roots of unity and cyclotomic polynomials, cyclic extensions, polynomial solvable by radicals, ruler and compass constructions Text Book: P. B. Bhattacharya, S. K. Jain and S. R. NagPaul, Basic Abstract Algebra, Cambridge University Press, Indian Edition, 1997 Chapter 15, 16, 17 and 18 complete Reference Books: 1. I. N. Herstein: Topics in algebra, Wiley Eastern Ltd., New Delhi, S. Lang: Algebra, 3 rd edition, Addison-Wesley, I. S. Luther and I.B.S. Passi: Algebra, Vol. I and Vol. II Narosa, New Delhi. 4. D. S. Malik, J. N. Mordeson and M. K. Sen: Fundamentals of Abstract Algebra, Mc Graw-Hill, and International Edition, S. K. Jain, A. Gunawardena and P. B. Bhattacharya: Basic Linear Algebra with MATLAB, Key College Publishing (Springer-Verlag), J. B. Fraleigh, a first curse in Abstract Algebra, Narosa Publications.

5 5 Semester I Paper II (A) Real Analysis- I Unit I Definition and existence of Riemann-Stieltjes integral, Properties of the integral, Integration and Differentiation, The fundamental theorem of calculus, Examples. Unit II Integration of vector valued functions. Rectifiable curve. Examples. Sequences and series of functions. Point wise and uniform convergence. Cauehy criterion for uniform convergence. Weierstrass M-test, uniform convergence and continuity, uniform convergence and Riemann-Stielljes integration. Examples. Unit III Uniform convergence and Differential, The Stone Weierstrass theorem, Examples. Power series, Abel s and Taylor s theorems, Uniqueness theorem for power series. Examples. Unit IV Functions of several variables, Linear transformations, Derivatives in an open subset of R n, Chain rule, Examples Unit V Partial derivations. Interchange of the order of differentiation, The inverse function theorem, The implicit function theorem Jacobins, Derivatives of higher order, Differentiation of integrals. Examples, (15 lecturer) Text Book: Walter Rudin, Principles of Mathematical Analysis, (3 rd Edition) McGraw Hill, Kogakusha Articles: 6.1 to 6.27, 7.1 to 7.18, 7.26, 7.27, 8.1 to 8.5, 9.1 to 9.21, 9.24 to 9.29, 9.38 to 9.42 Reference Books: 1. T. M. Apostol, mathematical Analysis, Narosa, New Delhi, J. C. Burkill and H. Burkill, A second course in Mathematical Analysis, Cambridge University Press, S. L. Lang, Analysis- I and II, Addison Wesley, 1969.

6 6 Semester - I Paper III (A) Topology - I Unit I Prerequisites: Partially ordered sets, Maximal and minimal elements, cardinality, special cardinals countable and uncountable sets, Axiom of choice continuum hypothesis, principle of inductions metric spaces, definition and Examples, continuous map, open sets properties of open sets, characterizations of continuity. Unit II Definition and examples of topological spaces, closed sets, closure of a set, properties of closure of sets, interior of a set and their properties, frontier of sets and its relation ship with closure and interior of sets neighbourhood of a point, Neighbourhood system, accumulation point and derived set. Unit III Bases and sub bases and related theorems new spaces from old. Sub spaces, continuous functions product spaces, weak topologies and related theorems open closed maps projection maps. Unit IV Evaluation map and related results Quotient spaces sequences in a topological space, Inadequacy of sequences, first countable spaces. Unit V Directed sets, nets, convergence of nets, cluster point, subnet, ultra net, filter, convergence of filters, ultra filters, fixed and free filters, results on these concepts. Textbook: Stephan Willard: General Topology, Addison Wesley (1970) Chapter 1 (Sec. 1.8 to 1.21 and Sec, 2.1 to 2.8) Chapter 2 (Complete), Chapter 3 (up to sec. 9.3) Chapter 4 (Complete) Reference Books: 1. Steen & J. Seecatch: Counter examples in Topology, Holt, Rinehart and instant, N, Y. (1970). 2. W. J. Pervin: Foundation of general Topology Academic press N.Y. 3. S. T. Hu. : Elements of general Topology, Holden. 4. James Munkres: Topology, A first course, Prentice Hall of India Pvt. Ltd.

7 7 Semester I Paper IV (A) - Complex Analysis - I Unit- I The Complex number system: The field of complex numbers, The complex plane, Rectangular and polar representation of complex numbers; Intrinsic function on the complex field; The Complex plane. Unit - II Metric spaces and Topology of C: Definition and examples of metric spaces; connectedness; sequence and completeness; compactness; continuity; Uniform convergence. (15 lecturer) Unit- III Elementary properties and examples of Analytic functions: Power series; The exponential function; Trigonometric and hyperbolic functions; Argument of nonzero complex number; Roots of unity; Branch of logarithm function. Analytic functions; cauchy Riemann Equations; Harmonic function; Unit - IV Analytic functions as a mapping; Mobius transformations; linear transformations; The point at infinity; Bilinear transformations, Complex Integration: power series representation of analytic functions; zeros of an analytic function. Unit V The index of a closed curve; cauchy s theorem and integral formula; Gaursat s Theorem; Singularities: Classification of singularities; Residues; The argument principle. Text Books: 1. John B. Conway; Functions of one complex variable, Narosa Publishing House, J. V. Deshpande; Complex Analysis, Tata McGraw- Hill Publishing Company Limited, Unit-I: Chapter-I: 2,3,4 in [1] & 1.3 & 1.4 in [2] Unit-II: Chapter II: 1,2,3,4,5,6 in [1] Unit III: Chapter VI: 6.1,6.2,6.3,6.4,6.5,6.6 in [2] & Chapter - VII: 7.1,7.2,7.3, in [2] Unit- IV: Chapter- III: 3 in [1] & Chapter IV: 2, 3 & 4 in [1] & Chapter 2: 2.1,2.2,2.3, in [2] Unit V: Chapter IV: 5 & 8 in [1] & Chapter V: 1,2 & 3 in [1]. References: 1. Herb Silverman; Complex Variables, Haughton Mifflin Company Boston, Ruel V. Churchill; Complex variables and applications, McGraw Hill Publishing Company 1990.

8 8 Semester I Paper V -a- (A)-Differential Equations - I Unit I Existence, uniqueness and Continuation of solutions: Introduction, Method of successive approximations for the initial value problem y 1 =f (x,y), y(x 0 )=y 0, The Lipschitz condition. Notation and Definitions, Peano s existence theorem, maximal and minimal solutions, continuation of solutions. Unit II Existence theorems for system of differential equations: Picard-Lindelof theorem, Peano s existence theorem, Dini s derivatives, differential inequalities. Unit III Integral Inequalities: Gronwall- Reid-Bellman inequality and its generalization, Applications: Zieburs theorem, Peron s criterion, Kamke s uniqueness theorem. Unit IV Linear systems: Introduction, superposition principle, preliminaries and Basic results, Properties of linear homogeneous system, Theorems on existence of a fundamental system of solutions of first order linear homogeneous system, Abel-Liouville formula. Unit V Adjoint system, Periodic linear system, Floquet s theorem and its consequences, Applications, Inhomogeneous linear systems, applications. Text Book: 1. E. A. Coddington: An Introduction to Ordinary Differential Equations. Prentice- Hall international, Inc. Englewood Cliffs (1961). Chapter 6: Article 4&5. 2. Shair Ahmad and M. Rama Mohana Rao: Theory of Ordinary Differential Equations with Applications in Biology and Engineering, Affiliated East-West Press (1999) Chapter 1: Article 1.1 to 1.5 Chapter 2: Article 2.1 to 2.3 References: 1. P. Hartman: Ordinary differential Equations, 2 nd edition, SIAM, (2002.) 2. W. T. Reid: Ordinary Differential Equations, John Wiley, New York, (1971). 3. E. A. Coddington and N. Levinson: Theory of Ordinary Differential Equations, McGraw-Hill, New York, (1955).

9 9 Semester I Semester - I PAPER V-b-(A): ADVANCED DISCRETE MATHEMATICS - I Unit I Formal Logic: Statements, symbolic representation, tautologies. Definitions and examples of semi groups and Monoids Semi groups and monoids: (15 Lecture) Unit- II Homomorphism of semigroups and monoids, congruence relation and quotient semigroups, Sub semigroups and submonoids, direct products, basic homomorphism theorem. (15 Lecture) Unit- III Lattices: Lattices as partially ordered sets, their properties, lattices as algebraic systems, sub lattices, direct products and homomorphism, some special lattices eg complete, complemented and distributive lattices. (15 Lecture) Unit- IV Boolean algebras: Boolean algebras as lattices, various Boolean identities, the switching algebra example, sub algebra, direct product and homomorphism, join-irreducible elements (15 Lecture) Unit- V Atoms and midterms, Boolean forms and their equivalence, midterm Boolean forms, (excluding free Boolean algebras), sum and products of canonical forms. Minimization of Boolean functions, applications of Boolean algebra to switching theory (using AND, ok and NOT gates), the Karnaugh Map method. (15 Lecture) Reference Books: 1. J. P. Tremblay and R. Manohar: Discrete Mathematical structures with Applications to Computer science, McGraw-Hill Book Co., Seymour Lepschutz: Finite Mathematics, McGraw-Hill, New York. 3. S. Wiitala: Discrete Mathematics - A Unified Approach, McGraw-Hill. 4. J. E. Hhopcroft and J.D. Ullman: Introduction to Automata Theory, Languages and Computation, Narosa, New Delhi. 5. C. L. Liu: Elements of discrete Mathematics, McGraw-Hill Book Co.

10 10 Semester I Paper V-c (A)- Differential Geometry of Manifolds - I Definition and examples of differentiable manifolds. Tangent spaces. Jacobean map. One parameter group of transformations. Lie derivatives. Immersions and imbedding. Distributions. Exterior algebra. Exterior derivative. Topological groups. Lie groups and lie algebras. Product of two Liegroups. One parameter subgroups and exponenitial maps. Examples of Liegroups. Homomorphism and Isomorphism. Lie transformation groups. General linear groups. Principal fibre ;bundle. Linear frame bundle. Associated fibre bundle. Vector bundle. Induced bundle. Bundle homomorphisms. Reference Books: 1. R.S. Mishra, A course in tensors with applications to Riemannian Geometry, Pothishala (pvt.) Ltd., R.S. Mishra, Structures on a differentiable manifold and their applications, Chandrama Prakashan, Allahabad, B. B. Sinha, An Introduction to Modern Differential Geometry, Kalyani Publishers, New Delhi, K. Yano and M. Kon, Structure of manifolds. World Scientific, 1984

RANI DURGAVATI UNIVERSITY, JABALPUR

RANI DURGAVATI UNIVERSITY, JABALPUR SYLLABUS OF M.A./M.Sc. MATHEMATICS SEMESTER SYSTEM Semester-II (Session 2012-13 and onwards) Syllabus opted by the board of studies in Mathematics, R. D. University